During creation of the Group Concept Inventory (GCI), we discovered that the initial question related to isomorphism adapted from Weber & Alcock (2004) (Are Q and Z isomorphic?) contained a number of hidden complexities related to student understanding of isomorphism. We developed two new questions to attempt to better parse apart student conceptions around the topic. One question targeted cardinality and the second structural sameness. The questions were piloted with group theory students across the country. We analyzed the data using a thematic analysis approach. Additionally, we conducted five follow-up interviews to further make sense of student reasoning related to isomorphism. In this presentation, we focus on student use of formal and informal definitions, their ability to explain the relationship between structural sameness and the formal definition, and their flexibility addressing non-familiar properties.